A use for integer quotients

[Michael Abbott]

Robin Garner <robin.garner at i.name.com> wrote in
news:Xns90E97C9849AD5robindotgarneratinam at 172.18.25.3: 

> "michael" <serrano at ozemail.com.au> wrote in
> news:JDa77.97061$Rr4.549007 at ozemail.com.au: 
> 
> In most maths the sets are made explicit, and operators are defined as 
> functions with specific domains.  There are standard embeddings of
> Integers into Rationals and Reals, and if you are working with real
> numbers, then 2 and 2.0 are generally taken to stand for the same
> thing.  But strictly speaking the Integer 2 and the Real Number 2 are
> different entities.  You would be hard pressed to find anyone writing
> the natural number 2 as 2.0. 

Indeed.  The domain and sets of operations are crucial, and it is important 
to keep integers and reals separate.  Of course we know that there is a 
injective mapping from integers to reals which preserves all sorts of nice 
properties the integers (but *not* all properties), but integers are not 
reals!

> 
> Since the Integers aren't a field, there isn't a generally accepted 
> function 
> 
>          / : Int x Int -> Int.
> 
> that agrees with the function 
> 
>           / : Real x Real -> Real
> 
> for all (a,b) where b is divisible by a.

However, since the integers are a "Euclidian domain", there is a function
    	divmod : Int x Int -> Int x Int
with the expected properties (where a/b is a plausible name for the first 
half of the result of this operation).